A317326 Multiples of 26 and odd numbers interleaved.

0, 1, 26, 3, 52, 5, 78, 7, 104, 9, 130, 11, 156, 13, 182, 15, 208, 17, 234, 19, 260, 21, 286, 23, 312, 25, 338, 27, 364, 29, 390, 31, 416, 33, 442, 35, 468, 37, 494, 39, 520, 41, 546, 43, 572, 45, 598, 47, 624, 49, 650, 51, 676, 53, 702, 55, 728, 57, 754, 59, 780, 61, 806, 63, 832, 65, 858, 67, 884, 69

Offset: 0

Author

Omar E. Pol, Jul 25 2018

Keywords

• nonn
• mult
• easy

Comments

a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).

Partial sums give the generalized 30-gonal numbers.

More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.

From Bruno Berselli, Jul 27 2018: (Start)

Also, this type of sequence is characterized by:

O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;

E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;

a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;

a(n) = (m/2)^((1 + (-1)^n)/2)*n;

a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A252994 and A005408 interleaved.

Column 26 of A195151.

Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16), A317313 (k=17), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).

Cf. A316729.

Programs

Julia

[13^div(1+(-1)^n,2)*n for n in 0:70] |> println # Bruno Berselli, Jul 28 2018

Mathematica

Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* Bruno Berselli, Jul 27 2018 *)

Formula

a(2*n) = 26*n, a(2*n+1) = 2*n + 1.

From Bruno Berselli, Jul 27 2018: (Start)

O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.

E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).

a(n) = -a(-n) = (7 + 6*(-1)^n)*n.

a(n) = 13^((1 + (-1)^n)/2)*n.

a(n) = 2*a(n-2) - a(n-4). (End)

Multiplicative with a(2^e) = 13*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023

Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(3-s)). - Amiram Eldar, Oct 26 2023